# AP Calc BC in a week - Part 5

## Introduction

Yay! Last day! Today is:

- 10 Sequences and Series

Let's start!

This section is long so I'm going to have to split it up quite a bit...

## 10 Sequences and Series

### What is a(n) (Infinite) Sequence?

An **infinite sequence** is a function which has the natural numbers as the domain (not including 0). It is denoted as \(a_n\).

An example of a sequence is \(a_n=\frac{1}{n}\), which results in the sequence \(\frac{1}{1},\frac{1}{2},\frac{1}{3}\cdots\frac{1}{n}\cdots\). \(\frac{1}{n}\) is often called the nth term of this sequence.

I suppose this is to contrast sequences from the other functions that we deal with, whose domain consist of all the real numbers.

A sequence \(a\) **converges to a finite number \(L\) if \(\lim_{n\to\infty}a_n=L\)**. If the limit does not exist then the sequence diverges.

tl;dr A sequence is a (function) list of numbers, and if the limit ( \(n\to\infty\) ) of its function is \(L\), then it converges to \(L\).

### What is a(n) (Infinite) Series?

An **infinite series** is the sum of an infinite sequence.
$$\sum_{k=1}^\infty a_k = a_1 + a_2 + a_3 + \cdots + a_n + \cdots$$
Each element in the sum is a *term*, with \(a_n\) being the nth term, like in the sequence.

The series \(\sum_{k=1}^\infty a_k\) can also be written as \(\sum a_k\) or \(\sum a_n\).

tl;dr Add all the numbers in a sequence = series

### Types of Series

**p-series** is the series of form $$\sum_{k=1}^\infty \frac{1}{k^p}$$, where \(p\) is a constant.

**Harmonic series** is the p-series where \(p=1\), ie $$\sum_{k=1}^\infty \frac{1}{k}$$

**Geometric series** is the sum of a first term \(a\) and terms of a common ratio \(r\) : $$\sum_{k=1}^\infty ar^{k-1}$$

### Convergence of a Series

If there exists a finite number \(S\) such that \(\lim_{n\to\infty}\sum_{k=1}^ne a_k = S\), then the inifinite series is said to converge to \(S\).

In this case, $$\sum_{k=1}^\infty = S$$.

tl;dr If you keep adding and it approaches a number, then the series converges to that number

There are a couple examples of proving that series are Divergent/convergent, but... I don't think I can reproduce these with an unseen problem, since these involve finding limits of summations, so... hopefully they can be reasoned through if they do come out.

### Theorems of Convergence of Divergence

#### If \(\sum a_n\) converges, then \(\lim_{n\to\infty}a_n=0\)

- By contraposition, this also means that if \(\lim_{k\to\infty}a_n\neq 0\), \(\sum a_n\) diverges
- The converse is not true; if the limit is zero, the summation may or may not converge (if you know your truth tables this won't be a problem)
- Note that this means that the terms go to 0, not the series

#### You can add a finite number of terms to a summation without affecting its convergence/divergence

- This means that \(\sum_{k=1}^\infty a_k\) and \(\sum_{k=m}^\infty a_k\) both converge or both diverge

#### A series can be multiplied by a nonzero constant without affecting its convergence/divergence

- This means that \(\sum_{k=1}^\infty a_k\) and \(c\sum_{k=1}^\infty a_k\) both converge or both diverge

#### If two series converge, the sum of the two series converges

- If \(\sum a_n\) and \(\sum b_n\) both converge, \(\sum (a_n+b_n)\) converges

#### If the terms of a convergent series are regrouped, the new series is also convergent

## Conclusion

Uhh okay I've had this buffer open for too long I need to publish this prematurely, plus the post will become way too long anyway.

Chapter 10 is longer than I expected... I'll have to break it up. Next time we'll be doing Tests for convergences, then after that we'll do the Power series.